Function transformation: what it means and how to use it
Describe vertical scale, horizontal scale, reflection, and translation of a graph.
Function transformation is one of 1 transformations formula in the precalculus section of this library, and it is used at high school level.
Why function transformation works
The numbers a and k act after the function has done its work, so they behave the way you expect: multiply the output, then raise it. The numbers b and h act before the function sees the input, so their effects run backwards. Dividing the input by a stretch factor is what makes the graph squeeze rather than stretch.
What each symbol means
$h,k$ shift the graph and $a,b$ scale or reflect it.
Function transformation: when it holds
$a\ne0$ and $b\ne0$ for a noncollapsed transformed graph; horizontal scale is $1/|b|$.
When it stops applying
The written order is not the order you may apply the steps in. Stretching then shifting vertically gives 2f(x) + 3, while shifting then stretching gives 2f(x) + 6, so the same two moves in the other sequence land on a different graph.
Function transformation: a worked example
$2f(3(x-1))-4$ shifts right $1$, scales horizontally by $1/3$, vertically by $2$, and down $4$.
The mistake to avoid
What people do: Students say the graph of f(3x) is stretched three times wider.
Why it goes wrong: Inside the parentheses everything is reversed. Multiplying the input by 3 means the function reaches any given value three times sooner, so the picture is squeezed to a third of its width.
Do this instead: Read horizontal changes as their opposites: the stretch factor is 1/|b|. So y = 2f(3(x − 1)) − 4 squeezes to one third of the width, doubles the height, moves right 1, and drops 4.
Function transformation: step by step
- Name the unknown, and the unit the answer has to come out in.
- Match the symbols to your values. $h,k$ shift the graph and $a,b$ scale or reflect it.
- Check the conditions before substituting. $a\ne0$ and $b\ne0$ for a noncollapsed transformed graph; horizontal scale is $1/|b|$.
- Substitute, keep exact values to the last line, then test the sign, size, and unit against a rough estimate — the check that catches most precalculus slips.
Where this formula fits
- Subject
- Precalculus formulas — 10 entries in this library
- Topic
- Transformations
- Level
- High school
Formulas are easiest to keep when they sit inside a method rather than on a list. Use the links below to see where function transformation comes from, to check a calculation against a tool, and to practise it until you can recall it without looking.
- Function Transformations lesson — the lesson behind this formula: predict shifts, reflections, and stretches from notation.
- Precalculus Calculator — check your substitution and the value it produces.
- Study precalculus — the subject guide that explains the ideas these formulas compress.
- Precalculus Practice — questions that make you retrieve the formula instead of recognising it.
- All 10 precalculus formulas — the full grouped reference, or the complete formula library.
Questions about function transformation
Why does x − 1 shift the graph to the right?
Because the input must grow by 1 before the function reaches the value it used to reach. Everything happens later, and later means further right on the graph.
What does a negative a do?
It flips the graph upside down across the horizontal axis, on top of any stretching its size causes. A negative b flips it left to right instead, across the vertical axis.
How do I read the shift when the inside is not factored?
Factor the b out first. In f(3x − 6) rewrite the inside as 3(x − 2), so the shift is 2 units rather than the 6 the unfactored version seems to suggest.
Which transformations preserve the shape of the graph?
Only the shifts and the reflections. Stretching by a or b changes proportions, so a circle can become an ellipse, while sliding and flipping move the picture without distorting it.